Numerical Integrators

2002 ◽  
pp. 23-46 ◽  
Author(s):  
Ernst Hairer ◽  
Gerhard Wanner ◽  
Christian Lubich
Keyword(s):  
Numerical Integrators

scholarly journals Multisymplectic formulation of fluid dynamics using the inverse map

2007 ◽  
Vol 463 (2086) ◽  
pp. 2671-2687 ◽  
Author(s):  
C.J Cotter ◽  
D.D Holm ◽  
P.E Hydon
Keyword(s):  
Fluid Dynamics ◽  
Conservation Laws ◽  
Fluid Particle ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Numerical Integrators ◽  
Fluid Particles ◽  
Infinite Set ◽  
Circulation Theorem

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.

scholarly journals ARBTools: A Tricubic Spline Interpolator for Three-Dimensional Scalar or Vector Fields

2019 ◽  
Author(s):  
Paul Walker ◽  
Ulrich Krohn ◽  
Carty David
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Spline Method ◽  
Particle Trajectories ◽  
The Core ◽  
Numerical Integrators ◽  
Data Points ◽  
System Requirements

ARBTools is a Python library containing a Lekien-Marsden type tricubic spline method for interpolating three-dimensional scalar or vector fields presented as a set of discrete data points on a regular cuboid grid. ARBTools was developed for simulations of magnetic molecular traps, in which the magnitude, gradient and vector components of a magnetic field are required. Numerical integrators for solving particle trajectories are included, but the core interpolator can be used for any scalar or vector field. The only additional system requirements are NumPy.

scholarly journals COMPARATIVE STUDY OF VARIATIONAL CHAOS INDICATORS AND ODEs' NUMERICAL INTEGRATORS

2012 ◽  
Vol 22 (10) ◽  
pp. 1230033 ◽  
Author(s):  
L. A. DARRIBA ◽  
N. P. MAFFIONE ◽  
P. M. CINCOTTA ◽  
C. M. GIORDANO
Keyword(s):  
Numerical Technique ◽  
Equations Of Motion ◽  
General System ◽  
Detailed Comparison ◽  
Dynamical Problem ◽  
Point Of View ◽  
The Other ◽  
Numerical Integrators ◽  
Fast Lyapunov Indicator ◽  
The Times

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon & Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI 2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.

scholarly journals Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions

2014 ◽  
Vol 55 (3) ◽  
pp. 705-732 ◽  
Author(s):  
David Cohen ◽  
Ludwig Gauckler ◽  
Ernst Hairer ◽  
Christian Lubich
Keyword(s):  
Hamiltonian Systems ◽  
Oscillatory Hamiltonian Systems ◽  
Numerical Integrators ◽  
Term Analysis

Numerical Integrators

2006 ◽  
pp. 27-50 ◽  
Author(s):  
Ernst Hairer ◽  
Gerhard Wanner ◽  
Christian Lubich
Keyword(s):  
Numerical Integrators

A Recursive Implementation Method With Implicit Integrator for Multibody Dynamics

2001 ◽  
Author(s):  
H. J. Cho ◽  
H. S. Ryu ◽  
D. S. Bae ◽  
J. H. Choi ◽  
B. Ross
Keyword(s):  
Equations Of Motion ◽  
Flexible Body ◽  
Differential Algebraic Equation ◽  
Dynamic Correction ◽  
Body Dynamics ◽  
Mode Method ◽  
Numerical Integrators ◽  
Flexible Body Dynamics ◽  
Body Concept ◽  
Implementation Method

Abstract A recursive implementation method for the equations of motion and kinematics is presented. Computational structure of the kinematic and dynamic equations is exploited to systematically implement a dynamic analysis program RecurDyn. A differential algebraic equation solution method with implicit numerical integrators is discussed. Virtual body concept is introduced for the flexible body dynamics. The accuracy of the flexible body solutions is estimated by an error measure and is improved by the dynamic correction mode method. Several examples are solved to demonstrate the efficiency of the proposed methods.

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